Properties

Label 139.1.b.a
Level 139
Weight 1
Character orbit 139.b
Self dual yes
Analytic conductor 0.069
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -139
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 139 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 139.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0693700367560\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.139.1
Artin image $S_3$
Artin field Galois closure of 3.1.139.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{4} - q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q + q^{4} - q^{5} - q^{7} + q^{9} - q^{11} - q^{13} + q^{16} - q^{20} - q^{28} - q^{29} - q^{31} + q^{35} + q^{36} + 2q^{37} + 2q^{41} - q^{44} - q^{45} + 2q^{47} - q^{52} + q^{55} - q^{63} + q^{64} + q^{65} - q^{67} - q^{71} + q^{77} - q^{79} - q^{80} + q^{81} - q^{83} - q^{89} + q^{91} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/139\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
138.1
0
0 0 1.00000 −1.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
139.b odd 2 1 CM by \(\Q(\sqrt{-139}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 139.1.b.a 1
3.b odd 2 1 1251.1.c.b 1
4.b odd 2 1 2224.1.e.a 1
5.b even 2 1 3475.1.c.b 1
5.c odd 4 2 3475.1.d.a 2
139.b odd 2 1 CM 139.1.b.a 1
417.d even 2 1 1251.1.c.b 1
556.d even 2 1 2224.1.e.a 1
695.d odd 2 1 3475.1.c.b 1
695.g even 4 2 3475.1.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
139.1.b.a 1 1.a even 1 1 trivial
139.1.b.a 1 139.b odd 2 1 CM
1251.1.c.b 1 3.b odd 2 1
1251.1.c.b 1 417.d even 2 1
2224.1.e.a 1 4.b odd 2 1
2224.1.e.a 1 556.d even 2 1
3475.1.c.b 1 5.b even 2 1
3475.1.c.b 1 695.d odd 2 1
3475.1.d.a 2 5.c odd 4 2
3475.1.d.a 2 695.g even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(139, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )( 1 + T ) \)
$3$ \( ( 1 - T )( 1 + T ) \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( 1 + T + T^{2} \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( ( 1 - T )( 1 + T ) \)
$23$ \( ( 1 - T )( 1 + T ) \)
$29$ \( 1 + T + T^{2} \)
$31$ \( 1 + T + T^{2} \)
$37$ \( ( 1 - T )^{2} \)
$41$ \( ( 1 - T )^{2} \)
$43$ \( ( 1 - T )( 1 + T ) \)
$47$ \( ( 1 - T )^{2} \)
$53$ \( ( 1 - T )( 1 + T ) \)
$59$ \( ( 1 - T )( 1 + T ) \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( 1 + T + T^{2} \)
$71$ \( 1 + T + T^{2} \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( 1 + T + T^{2} \)
$83$ \( 1 + T + T^{2} \)
$89$ \( 1 + T + T^{2} \)
$97$ \( ( 1 - T )( 1 + T ) \)
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